Partition function of modified Ising model

[LCSphysicist]

Homework Statement

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Relevant Equations

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$$H = - J ( \sum_{i = odd}) \sigma_i \sigma_{i+1} - \mu H ( \sum_{i} \sigma_i ) $$
So basically, my idea was to separate the particles in this way::
$N_{\uparrow}$ is the number of up spin particles

$N_{\downarrow}$ "" down spin particles

$N_1$ is the number of pairs of particles close to each other with spin up

$N_2$ "" with spin down

$N_3$ "" with spin antiparallel
Therefore
$$\hat{H} = - J (N_1 + N_2 - N_3 ) - \mu H (N_{\uparrow} - N_{\downarrow})$$
Subject to the following constraints:
$$\frac{N}{2} = N_1 + N_2 + N_3$$

$$N_{\uparrow} = 2N_1 + N_3$$

$$N_{\downarrow} = 2N_2 + N_3$$

$$N = N_{\uparrow} + N_{\downarrow}$$
But, even make this clarifications, i can't see how to find the partition! If we substitute the above expressions on $\sum e^{-\hat{H}/kT}$, we will find Z as a function of, for example, $Z=Z(N_{\uparrow},N_3)$. But i don't know how to evaluate such a sum! I mean, it can't factoriz in the sum of two expoents, because both arguments are constrained! (We can't have $N_3 = N/2, N_{\uparrow} = 0$, for example)

 

[LCSphysicist]

Done!